Final answer:
If S is a finite set with cardinality n, the cardinality of its powerset P(S) is 2^n.
Step-by-step explanation:
The cardinality of a set is a measure of the number of elements in the set. The cardinality of a set S, denoted as |S|, is a non-negative integer.
If S is a finite set and the cardinality of S is n, then we can prove that the cardinality of the powerset of S, denoted as P(S), is equal to 2^n.
To prove this, we can use the concept of binary representation. Each element in S can be represented by a 0 or a 1 in a binary string. For each element, if it is included in a subset, we assign a 1 to its position in the binary string; otherwise, we assign a 0.
Since there are n elements in S, there are 2^n possible binary strings of length n. Each binary string corresponds to a unique subset of S. Therefore, the cardinality of P(S) is equal to 2^n.